I would like to know if series of the form $ \sum_{n>0}a_{n}\dfrac{q^{tn}}{n^{z(1-t)}} $ where $ t\in(0,1) $ , $ a_{n} $ is the $ n $ -th Fourier coefficient of a cusp form of a given weight $k $ and level $ N $ and $q=e^{2i\pi z} $ for $ z $ in the upper half plane of complex numbers with positive imaginary part have been considered so far. If so, is the value of (the derivative of) such a series for $ t=1/2 $ of interest ? Can an analogue of the Rankin-Selberg convolution be defined for such series ?
Continuous transformation of a q-expansion into a Dirichlet series
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number-theory
modular-forms
dirichlet-series
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3Why would these appear? In what context are you arriving at expressions of this form? – 2017-02-07
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0I just thought that instead of attaching an L-function to a modular form, which in some sense boils down to jumping discretely from the first to the second, it might be interesting to deform the first continuously into the second. – 2017-02-08
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0You need to study the basics of zeta functions, for example proofs of the functional equation. $F(s) = \sum_{n=1}^\infty a_n n^{-s}$, $f(q,z) = \sum_{n=1}^\infty a_n q^n n^{-z}$. $\Gamma(s) F(s) = \int_0^\infty x^{s-1} f(e^{-x},0)dx$ and $$\Gamma(s) F(s+z) = \int_0^\infty x^{s-1} f(e^{-x},z)dx$$ With $q^{t n}$ it works the same after a change of variable $y = tx$. Rankin-Selberg convolution works well only for automorphic functions, otherwise it doesn't tell us anything. @mixedmath – 2017-03-20